A $0\mathrm{.}500$ kg sphere moving with a velocity given by $(2\mathrm{.}00$hat$(i)-3\mathrm{.}20$hat$(j)+1\mathrm{.}00$hat$(k))\frac{m}{s}$ strikes another sphere of mass $1\mathrm{.}50$ kg moving with an initial velocity of $(-1\mathrm{.}00$hat$(i)+2\mathrm{.}00$hat$(j)-3\mathrm{.}30$hat$(k))\frac{m}{s}.$

$($a$)$ The velocity of the $0\mathrm{.}500$ kg sphere after the collision is $(-0\mathrm{.}80$hat$(i)+3\mathrm{.}00$hat$(j)-8\mathrm{.}00$hat$(k))\frac{m}{s}.$ Find the final velocity of the $1\mathrm{.}50$ kg sphere.

vec$(v)=(-0\mathrm{.}067$hat$(i)-0\mathrm{.}067$hat$(j)-0\mathrm{.}30$hat$(k))\frac{m}{s}$

Identify the kind of collision $($elastic$,$ inelastic, or perfectly inelastic$).$

$$

$($b$)$ Now assume the velocity of the $0\mathrm{.}500$ kg sphere after the collision is $(-0\mathrm{.}250$hat$(i)+0\mathrm{.}700$hat$(j)-2\mathrm{.}23$hat$(k))\frac{m}{s}.$ Find the final velocity of the $1\mathrm{.}50$ kg sphere.

vec$(v)=\frac{m}{s}$

Identify the kind of collision.

elastic

inelastic

perfectly inelastic

$($c$)$ Take the velocity of the $0\mathrm{.}500$ kg sphere after the collision as $(-1\mathrm{.}00$vec$(i)+2\mathrm{.}80$vec$(j)+$avec$(k))\frac{m}{s}.$ Find the value of a and the velocity of the $1\mathrm{.}50$ kg sphere after an elastic collision. $($Two values of a are possible, a positive value and a negative value. Report each with their corresponding final velocities.$)$

a $($positive value$)$

Your response differs from the correct answer by more than $10\%.$ Double check your calculations.

a $($negative $$ Your response is within $10\%$ of the correct value. This may be due to roundoff error, or you could have a mistake in your value$)$ calculation. Carry out all intermediate results to at least four$-$digit accuracy to minimize roundoff error. Correct ans please